### AP Calculus BC Study Guide: Comparison Tests for Convergence

#### Introduction

Hello, calculus enthusiasts! 🧮 Welcome to the fantastical world of Infinite Sequences and Series in AP Calculus BC. Today, we are diving deep into the Comparison Tests for Convergence. Think of these tests as the ultimate showdown where series compete to see which ones converge or diverge. They're the gladiator games of the mathematical world! 🏛️⚔️

#### Comparison Test Theorems

Comparison tests are your trusty sidekicks when dealing with series that are too gnarly to handle directly. Imagine they're like the friends who help you find your way out of a maze. There are two main heroes here: the Direct Comparison Test and the Limit Comparison Test. 🎭

**Direct Comparison Test**

The Direct Comparison Test is like comparing a giant to a mini-giant—you have two series, ∑an and ∑bn, where both an and bn are non-negative. If the terms of one series (an) are always less than or equal to the terms of another series (bn), then:

- If ∑bn converges, then ∑an also converges. It's like saying if the bigger giant can sit down, so can the mini-giant.
- If ∑an diverges, then ∑bn also diverges. It's like saying if the mini-giant can't reach the ceiling, the giant definitely can't.

Visualize an as the smaller underdog and bn as the mightier champion. If bn makes it to convergence town, an is sure to follow. Conversely, if an fails to diverge town, bn won't get there either.

**Limit Comparison Test**

Sometimes you can't directly compare series; they're like two best friends who took slightly different routes to the same place. In comes the Limit Comparison Test, which says if you have two series ∑an and ∑bn (both positive), and you take the limit as n approaches infinity of the ratio an/bn and get a positive finite number, both series either converge or diverge together.

Think of an and bn as two marathon runners. If you check them out at any time and find they're running at the same pace, they'll end the race together—either both finishing (converging) or both dropping out (diverging). 🚴♂️

#### Breaking Down the Theorems with Examples

Let's walk through some examples to see these towering math heroes in action!

**Example 1: Direct Comparison Test**

Determine if this series converges:

∑n=1∞ (5/(2n² + 4n + 3))

Without help, solving this series feels like trying to climb Mount Everest with flip-flops. Instead, compare this series to ∑n=1∞ (5/n²).

Notice how 2n² + 4n + 3 ≤ n² (as 2n² ≥ n²). Therefore:

5/(2n² + 4n + 3) ≤ 5/n².

We know ∑5/n² converges because it's a p-series with p = 2, which converges. According to our Direct Comparison Test, our series also converges. Victory! 🥇

**Example 2: Limit Comparison Test**

Determine if this series converges:

∑n=1∞ (1/(3ⁿ - n))

This time, try comparing it with another series like ∑n=1∞ (1/3ⁿ).

Calculate the limit of their ratio:

lim n→∞ [(1/(3ⁿ - n)) / (1/3ⁿ)] = lim n→∞ [3ⁿ / (3ⁿ - n)] = 1.

Since the limit is a positive finite number (1), both series either converge or diverge together. We know ∑n=1∞ (1/3ⁿ) converges because it's a geometric series. By the Limit Comparison Test, our original series also converges. Huzzah! 🎉

#### Practice Problems

Now it's your turn, math warriors! Sharpen your pencils and your wits:

- Determine if the series ∑n=1∞ ((2n² + 3n) / √(5 + n⁵)) converges or diverges.
- Examine if ∑n=1∞ ((n² + 3) / (n³ + 1)) converges or diverges.
- Check if the series ∑n=1∞ (sin(n) / n³) converges or diverges.

**Solution Hints**:

**Problem 1**:

Compare with ∑n=1∞ (2/√n). Use L'Hopital's Rule for verification.

**Problem 2**:

Compare with the harmonic series ∑n=1∞ (1/n).

**Problem 3**:

Use direct comparison with ∑n=1∞ (1/n³) as sin(n) ≤ 1.

#### Conclusion

Congratulations, you've explored the mystical realms of Comparison Tests in AP Calculus BC! Just remember, whether you're facing a giant or a mini-giant, these tests will guide you to victory. So arm yourself with these tools, practice, and soon you'll be the hero of convergence and divergence battles! 🚀🦸♀️

Happy calculating, and may the limits always be in your favor!